reserve x,y for set,
  n,m for Nat,
  r,s for Real;

theorem Th6:
  for f be FinSequence, n,m be Nat holds
  n in dom f & m in Seg n implies (f|n).m = f.m & m in dom f
proof
  let f be FinSequence, n,m be Nat;
  assume that
A1: n in dom f and
A2: m in Seg n;
  dom f = Seg len f & n<=len f by A1,FINSEQ_1:def 3,FINSEQ_3:25;
  then
A4: Seg n c= dom f by FINSEQ_1:5;
  then Seg n = dom f /\ Seg n by XBOOLE_1:28
    .= dom(f|n) by RELAT_1:61;
  hence thesis by A2,A4,FUNCT_1:47;
end;
